College Physics

Hugh D. Young

Chapter 21

Electromagnetic Induction - all with Video Answers

Educators

Chapter Questions

Problem 1

A circular area with a radius of 6.50 $\mathrm{cm}$ lies in the $x$ -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field $B=0.230 \mathrm{T}$ that points (a) in the $+z$ direction? (b) at an angle of $53.1^{\circ}$ from the $+z$ direction? (c) in the $+y$ direction?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 2

The magnetic field $\vec{\boldsymbol{B}}$ in a certain region is $0.128 \mathrm{T},$ and its direction is that of the $+z$ axis in Figure 21.47 (a) What is the magnetic flux across the surface abcd in the figure? (b) What is the magnetic flux across the surface befc? (c) What is the magnetic flux across the surface aefd? (d) What is the net flux through all five surfaces that enclose the shaded volume?

Ryan Hood

Numerade Educator

Problem 3

An open plastic soda bottle with an opening diameter of 2.5 $\mathrm{cm}$ is placed on a table. A uniform 1.75 T magnetic field directed upward and oriented $25^{\circ}$ from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 4

A single loop of wire with an area of 0.0900 $\mathrm{m}^{2}$ is in a uniform magnetic field that has an initial value of 3.80 $\mathrm{T}$ , is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 $\mathrm{T} / \mathrm{s}$ . (a) What emf is induced in this loop? (b) If the loop has a resistance of $0.600 \Omega,$ find the current induced in the loop.

Averell Hause

Carnegie Mellon University

Problem 5

A coil of wire with 200 circular turns of radius 3.00 $\mathrm{cm}$ is in a uniform magnetic field along the axis of the coil. The coil has $R=40.0 \Omega$ . At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 $\mathrm{A}$ in the coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 6

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 $\mathrm{cm}^{2}$ is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth's magnetic field at the location of the laboratory is $6.0 \times 10^{-5} \mathrm{T.}$ (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Averell Hause

Carnegie Mellon University

Problem 7

A closely wound rectangular coil of 80 turns has dimensions of 25.0 $\mathrm{cm}$ by 40.0 $\mathrm{cm}$ . The plane of the coil is rotated from a position where it makes an angle of $37.0^{\circ}$ with a magnetic field of 1.10 $\mathrm{T}$ to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 8

A very long, straight solenoid with a cross-sectional area of 6.00 $\mathrm{cm}^{2}$ is wound with 40 turns of wire per centimeter, and the windings carry a current of 0.250 A. A secondary winding of 2 turns encircles the solenoid at its center. When the primary circuit is opened, the magnetic field of the solenoid becomes zero in 0.0500 s. What is the average induced emf in the secondary coil?

Averell Hause

Carnegie Mellon University

Problem 9

A $30.0 \mathrm{cm} \times 60.0 \mathrm{cm} \mathrm{rec}-$ tangular circuit containing a a 15$\Omega$ resistor is perpendicular to a uniform magnetic field that starts out at 2.65 $\mathrm{T}$ and steadily decreases at 0.25 $\mathrm{T} / \mathrm{s}$ . (See Figure $21.48 . )$ While this field is changing, what does the ammeter read?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 10

$\bullet$ A circular loop of wire with a radius of 12.0 $\mathrm{cm}$ is lying flat on a tabletop. A magnetic field of 1.5 $\mathrm{T}$ is directed vertically upward through the loop (Figure 21.49 ). (a) If the loop is removed from the field region in a time interval of 2.0 $\mathrm{ms}$ , find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Averell Hause

Carnegie Mellon University

Problem 11

A flat, square coil with 15 turns has sides of length 0.120 $\mathrm{m}$ . The coil rotates in a magnetic field of 0.0250 $\mathrm{T}$ (a) What is the angular velocity of the coil if the maximum emf produced is 20.0 $\mathrm{mV}^{\prime} ?$ (Hint: Look at the motional emf induced across the ends of the segments of the coil.) (b) What is the average emf at this angular velocity?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 12

A cardboard tube is wrapped with two windings of insulated wire, as shown in Figure $21.50 .$ Is the induced current in the resistor $R$ directed from left to right or from right to left in the following circumstances? The current in winding $A$ is directed (a) from $a$ to $b$ and is increasing, (b) from $b$ to $a$ and is decreasing, $(c)$ from $b$ to $a$ and is increasing, and (d) from $b$ to $a$ and is constant.

Averell Hause

Carnegie Mellon University

Problem 13

A circular loop of wire is in a spatially uniform magnetic field, as shown in Figure $21.51 .$ The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) $B$ is increasing; (b) $B$ is decreasing; (c) $B$ is constant with a value of $B_{0}$ . Explain your reasoning.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 14

Using Lenz's law, determine the direction of the current in resistor $a b$ of Figure 21.52 when (a) switch $S$ is opened after having been closed for several minutes; (b) coil $B$ is brought closer to coil $A$ with the switch closed; (c) the resistance of $R$ is decreased while the switch remains closed.

Averell Hause

Carnegie Mellon University

Problem 15

A solenoid carrying a current $I$ is moving toward a metal ring, as shown in Figure $21.53 .$ In what direction, clockwise or counterclockwise (as seen from the solenoid) is a current induced in the ring? In what direction will the induced current be if the solenoid now stops moving toward the ring, but the current in it begins to decrease?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 16

A metal bar is pulled to the right perpendicular to a uniform magnetic field. The bar rides on parallel metal rails connected through a resistor, as shown in Figure $21.54,$ so the apparatus makes a complete circuit. Find the direction of the current induced in the circuit in two ways: (a) by looking at the magnetic force on the charges in the moving bar and (b) using Lenz's law.

Averell Hause

Carnegie Mellon University

Problem 17

Two closed loops $A$ and $C$ are close to a long wire carrying a current $I$ . (See Figure $21.55 . )$ Find the direction (clockwise or counterclockwise) of the current induced in each of these loops if $I$ is steadily increasing.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 18

A bar magnet is held above a circular loop of wire as shown in Figure 21.56 . Find the direction (clockwise or counterclockwise, as viewed from below the loop) of the current induced in this loop in each of the following cases. (a) The loop is dropped. (b) The magnet is dropped. (c) Both the loop and magnet are dropped at the same instant.

Averell Hause

Carnegie Mellon University

Problem 19

The current in Figure. 21.57 obeys the equation $I e=I_{0} e^{-2 b t}$ , where $b>0 .$ Find the direction (clockwise or counterclockwise) of the current induced in the round coil for $t>0$ .

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 20

A bar magnet is close to a metal loop. When this magnet is suddenly moved to the left away from the loop, as shown in Figure $21.58,$ a counterclockwise current is induced in the coil, as viewed by an observer looking through the coil toward the magnet. Identify the north and south poles of the magnet.

Averell Hause

Carnegie Mellon University

Problem 21

A very thin 15.0 $\mathrm{cm}$ copper bar is aligned horizontally along the east-west direction. If it moves horizontally from south to north at 11.5 $\mathrm{m} / \mathrm{s}$ in a vertically upward magnetic field of $1.22 \mathrm{T},$ (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 22

When a thin 12.0 $\mathrm{cm}$ iron rod moves with a constant velocity of 4.50 $\mathrm{m} / \mathrm{s}$ perpendicular to the rod in the direction shown in Figure 21.59 , the induced emf across its ends is measured to be 0.450 $\mathrm{V}$ . What is the magnitude of the magnetic field? (b) Which point is at a higher potential, $a$ or $b ?(\mathrm{c})$ If the bar is rotated clockwise by $90^{\circ}$ in the plane of the paper, but keeps the same velocity, what is the potential difference induced across its ends?

Averell Hause

Carnegie Mellon University

Problem 23

You're driving at 95 $\mathrm{km} / \mathrm{h}$ in a direction $35^{\circ}$ east of north, in a region where the earth's magnetic field of $5.5 \times 10^{-5} \mathrm{T}$ is horizontal and points due north. If your car measures 1.5 $\mathrm{m}$ from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 24

A 1.41 $\mathrm{m}$ bar moves through a uniform, 1.20 $\mathrm{T}$ magnetic field with a speed of 2.50 $\mathrm{m} / \mathrm{s}$ (Figure $21.60 ) .$ In each case, find the emf induced between the ends of this bar and identify which, if any, end $(a$ or $b)$ is at the higher potential. The bar moves in the direction of $(a)$ the $+x$ -axis; $($ b) the $-y$ -axis; (c) the $+z$ -axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with $b$ at a higher potential than $a,$ and what is this maximum emf?

Averell Hause

Carnegie Mellon University

Problem 25

The conducting rod $a b$ shown in Figure 21.61 makes frictionless contact with metal rails $c a$ and $d b .$ The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed 7.50 $\mathrm{m} / \mathrm{s}$ . (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit $a b d c$ is a constant 1.50$\Omega$ , find the magnitude and direction of the force required to keep the rod moving to the right with a constant speed of 7.50 $\mathrm{m} / \mathrm{s}$ .

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 26

Measuring blood flow. Blood contains positive and negative ions and therefore is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter $d$ of the vessel moving with speed $v$ .(See Figure $21.62 . )$ (a) If the blood vessel is placed in a magnetic field $B$ perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is $\mathcal{E}=v B d .$ (b) If you expect that the blood will be flowing at 15 $\mathrm{cm} / \mathrm{s}$ for a vessel 5.0 $\mathrm{mm}$ in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 $\mathrm{mV} ?$ (c) Show that the volume rate of flow $(R)$ of theblood is equal to $R=\pi \mathcal{E} d / 4 B .$ (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential $\mathcal{E}$ must be made directly across the vessel.)

Averell Hause

Carnegie Mellon University

Problem 27

A toroidal solenoid has a mean radius of 10.0 $\mathrm{cm}$ and a cross-sectional area of 4.00 $\mathrm{cm}^{2}$ and is wound uniformly with 100 turns. A second coil with 500 turns is wound uniformly on top of the first. What is the mutual inductance of these coils?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 28

A 10.0 -cm-long solenoid of diameter 0.400 $\mathrm{cm}$ is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils?

Averell Hause

Carnegie Mellon University

Problem 29

Two coils are wound around the same cylindrical form, like the coils in Example $21.8 .$ When the current in the first coil is decreasing at a rate of $0.242 \mathrm{A} / \mathrm{s},$ the induced emf in the second coil has magnitude 1.65 $\mathrm{mV}$ . (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the average magnetic flux through each turn when the current in the first coil equals 1.20 $\mathrm{A} ?(\mathrm{c})$ If the current in the second coil increases at a rate of $0.360 \mathrm{A} / \mathrm{s},$ what is the magnitude of the induced emf in the first coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 30

One solenoid is centered inside another. The outer one has a length of 50.0 $\mathrm{cm}$ and contains 6750 coils, while the coaxial inner solenoid is 3.0 $\mathrm{cm}$ long and 0.120 $\mathrm{cm}$ in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 $\mathrm{A} / \mathrm{s}$ (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid.

Averell Hause

Carnegie Mellon University

Problem 31

Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 $\mathrm{Wb}$ (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is $2.54 \mathrm{A},$ what is the average flux through each turn of solenoid 1 ?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 32

A 4.5 $\mathrm{mH}$ toroidal inductor has 125 identical equally spaced coils. (a) If it carries an 11.5 A current, how much magnetic flux passes through each of its coils? (b) If the potential difference across its ends is $1.16 \mathrm{V},$ at what rate is the current in it changing?

Averell Hause

Carnegie Mellon University

Problem 33

At the instant when the current in an inductor is increasing at a rate of 0.0640 $\mathrm{A} / \mathrm{s}$ , the magnitude of the self-induced emf is 0.0160 $\mathrm{V} .$ What is the inductance of the inductor?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 34

An inductor has inductance of 0.260 $\mathrm{H}$ and carries a current that is decreasing at a uniform rate of 18.0 $\mathrm{mA} / \mathrm{s} .$ Find the self-induced emf in this inductor.

Averell Hause

Carnegie Mellon University

Problem 35

A 2.50 $\mathrm{mH}$ toroidal solenoid has an average radius of 6.00 $\mathrm{cm}$ and a cross-sectional area of 2.00 $\mathrm{cm}^{2} .$ (a) How many coils does it have? (Make the same assumption as in Example $21.10 .$ ) (b) At what rate must the current through it change so that a potential difference of 2.00 $\mathrm{V}$ is developed across its ends?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 36

Self-inductance of a solenoid. A long, straight solenoid has $N$ turns, a uniform cross-sectional area $A,$ and length $l .$ Use the definition of self-inductance expressed by Equation 21.13 to show that the inductance of this solenoid is given approximately by the equation $L=\mu_{0} A N^{2} / l .$ Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because $B$ is actually smaller at the ends than at the center of the solenoid. For this reason, your answer is actually an upper limit on the inductance.)

Averell Hause

Carnegie Mellon University

Problem 37

When the current in a toroidal solenoid is changing at a rate of 0.0260 A/s, the magnitude of the induced emf is 12.6 $\mathrm{mV}$ . When the current equals $1.40 \mathrm{A},$ the average flux through each turn of the solenoid is 0.00285 $\mathrm{Wb}$ . How many turns does the solenoid have?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 38

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is $25.0 \mathrm{V},$ (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a $125 . \Omega$ resistor?

Averell Hause

Carnegie Mellon University

Problem 39

Off to Europe! You plan to take your hair blower to Europe, where the electrical outlets put out 240 $\mathrm{V}$ instead of the 120 $\mathrm{V}$ seen in the United States. The blower puts out 1600 $\mathrm{W}$ at 120 $\mathrm{V}$ . (a) What could you do to operate your blower via the 240 $\mathrm{V}$ line in Europe? (b) What current will your blower draw from a European outlet? (c) What resistance will your blower appear to have when operated at 240 $\mathrm{V} ?$

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 40

You need a transformer that will draw 15 $\mathrm{W}$ of power from a 220 $\mathrm{V}$ power line, stepping the voltage down to $6.0 \mathrm{V}(\mathrm{rms}),$ (a) What will be the current in the secondary coil? (b) What should be the resistance of the secondary circuit? (c) What will be the equivalent resistance of the input circuit?

Averell Hause

Carnegie Mellon University

Problem 41

A step-up transformer. A transformer connected to a 120 $\mathrm{V}$ (rms) ac line is to supply $13,000 \mathrm{V}$ (rms) for a neon sign. To reduce the shock hazard, a fuse is to be inserted in the primary circuit and is to blow when the rms current in the secondary circuit exceeds 8.50 $\mathrm{mA}$ (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 $\mathrm{mA}$ ? (c) What current rating should the fuse in the primary circuit have?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 42

An air-filled toroidal solenoid has a mean radius of 15.0 $\mathrm{cm}$ and a cross-sectional area of 5.00 $\mathrm{cm}^{2} .$ When the current is $12.0 \mathrm{A},$ the energy stored is 0.390 $\mathrm{J} .$ How many turns does the winding have?

Averell Hause

Carnegie Mellon University

Problem 43

Energy in a typical inductor. (a) How much energy is stored in a 10.2 $\mathrm{mH}$ inductor carrying a 1.15 A current? (b) How much current would such an inductor have to carry to store 1.0 $\mathrm{J}$ of energy? Is this a reasonable amount of current for ordinary laboratory circuit elements?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 44

(a) What would have to be the self-inductance of a solenoid for it to store 10.0 $\mathrm{J}$ of energy when a 1.50 $\mathrm{A}$ current runs through it? (b) If this solenoid's cross-sectional diameter is $4.00 \mathrm{cm},$ and if you could wrap its coils to a density of 10 coils/mm, how long would the solenoid be? (See problem $36 .$) Is this a realistic length for ordinary laboratory use?

Averell Hause

Carnegie Mellon University

Problem 45

A solenoid 25.0 $\mathrm{cm}$ long and with a cross-sectional area of 0.500 $\mathrm{cm}^{2}$ contains 400 turns of wire and carries a current of 80.0 A. Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coil's magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 46

Large inductors have been proposed as energy-storage devices. (a) How much electrical energy is converted to light and thermal energy by a 200 $\mathrm{W}$ lightbulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is $80.0 \mathrm{A},$ what is the inductance?

Averell Hause

Carnegie Mellon University

Problem 47

When a certain inductor carries a current $I,$ it stores 3.0 $\mathrm{mJ}$ of magnetic energy. How much current (in terms of $I )$ would it have to carry to store 9.0 $\mathrm{mJ}$ of energy?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 48

A $12.0 \mathrm{~V}$ dc battery having no appreciable internal resistance, a $150.0 \Omega$ resistor, an $11.0 \mathrm{mH}$ inductor, and an open switch are all connected in series. After the switch is closed, what are (a) the time constant for this circuit, (b) the maximum current that flows through it, (c) the current $73.3 \mu$ s after the switch is closed, and (d) the maximum energy stored in the inductor?

Averell Hause

Carnegie Mellon University

Problem 49

An inductor with an inductance of 2.50 $\mathrm{H}$ and a resistor with a resistance of 8.00$\Omega$ are connected to the terminals of a battery with an emf of 6.00 $\mathrm{V}$ and negligible internal resistance. Find (a) the initial rate of increase of the current in the circuit, (b) the initial potential difference across the inductor, (c) the current 0.313 s after the circuit is closed, and (d) the maximum current.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 50

In Figure $21.63, \quad$ both switches $S_{1}$ and $S_{2}$ are initially open. $S_{1}$ is then closed and left closed until a constant current is established. Then $S_{2}$ is closed just as $S_{1}$ is opened, taking the battery out of the circuit. (a) What is the initial current in the resistor just after $S_{2}$ is closed and $S_{1}$ is opened? (b) What is the time constant of the circuit? (c) What is the current in the resistor after a large number of time constants have elapsed?

Averell Hause

Carnegie Mellon University

Problem 51

In the circuit shown in Figure $21.64,$ the battery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter $(A)$ and voltmeters $\left(V_{1}$ and $V_{2}\right)$ (a) the instant after the switch is closed; (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were 24.0 $\mathrm{mH}$ instead?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 52

A 35.0 $\mathrm{V}$ battery with negligible internal resistance, a 50.0 $\mathrm{V}$ resistor, and a 1.25 $\mathrm{mH}$ inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach one-half of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

Averell Hause

Carnegie Mellon University

Problem 53

A 1.50 $\mathrm{mH}$ inductor is connected in series with a dc battery of negligible internal resistance, a 0.750 $\mathrm{k} \Omega$ resistor, and an open switch. How long after the switch is closed will it take for (a) the current in the circuit to reach half of its maximum value, (b) the energy stored in the inductor to reach half of its maximum value? (Hint: You will have to solve an exponential equation.)

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 54

A 12.0$\mu \mathrm{F}$ capacitor and a 5.25 $\mathrm{mH}$ inductor are connected in series with an open switch. The capacitor is initially charged to 6.20$\mu \mathrm{C}$ . What is the angular frequency of the charge oscillations in the capacitor after the switch is closed?

Averell Hause

Carnegie Mellon University

Problem 55

A 5.00$\mu \mathrm{F}$ capacitor is initially charged to a potential of 16.0 $\mathrm{V}$ . It is then connected in series with a 3.75 $\mathrm{mH}$ inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 56

A 15.0$\mu \mathrm{F}$ capacitor is charged to 175$\mu \mathrm{C}$ and then connected across the ends of a 5.00 $\mathrm{mH}$ inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?

Averell Hause

Carnegie Mellon University

Problem 57

An inductor is connected to the terminals of a battery that has an emf of 12.0 $\mathrm{V}$ and negligible internal resistance. The current is 4.86 $\mathrm{mA}$ at 0.725 $\mathrm{ms}$ after the connection is completed. After a long time the current is 6.45 $\mathrm{mA}$ . What are (a) the resistance $R$ of the inductor and (b) the inductance $L$ of the inductor?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 58

A rectangular circuit is moved at a constant velocity of 3.0 $\mathrm{m} / \mathrm{s}$ into, through, and then out of a uniform 1.25 T magnetic field, as shown in Figure $21.65 .$ The magnetic field region is considerably wider than 50.0 $\mathrm{cm} .$ Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field, (b) totally within the magnetic field, but still moving, and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

Averell Hause

Carnegie Mellon University

Problem 59

The rectangular loop in Figure $21.66,$ with area $A$ and resistance $R,$ rotates at uniform angular velocity $\omega$ about the $y$ axis. The loop lies in a uniform magnetic field $\vec{B}$ in the direction of the $x$ axis. Sketch graphs of the following quantities, as functions of time, letting $t=0$ when the loop is in the position shown in the figure: (a) the magnetic flux through the loop, (b) the rate of change of flux with respect to time, (c) the induced emf in the loop, (d) the induced emf if the angular velocity is doubled.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 60

A flexible circular loop 6.50 $\mathrm{cm}$ in diameter lies in a magnetic field with magnitude 0.950 T, directed into the plane of the page as shown in Figure $21.67 .$ The loop is pulled at the points indicated by the arrows, forming a loop of zero area in 0.250 s. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in $R :$ from $a$ to $b$ or from $b$ to $a$ ? Explain your reasoning.

Averell Hause

Carnegie Mellon University

Problem 61

An electromagnetic car alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of 3500 $\mathrm{Hz}$ . To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That's why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 $\mathrm{V}$ (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store 0.0160 $\mathrm{J}$ of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 62

In the circuit shown in Figure $21.68, S_{1}$ has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, $S_{2}$ is closed and $S_{1}$ is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

Averell Hause

Carnegie Mellon University

Problem 63

Consider the circuit in Figure 21.69 . (a) Just after the switch is closed, what is the current through each of the resistors? (b) After the switch has been closed a long time, what is the current through each resistor? (c) After $S$ has been closed a long time, it is opened again. Just after it is opened, what is the current through the 20.0$\Omega$ resistor?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 64

In the diagram of TMS shown in Figure $21.70,$ a current pulse increases to a peak and then decreases to zero in the direction shown in the stimulating coil. What will be the direction $(1$ or 2$)$ of the induced current (dotted line) in the brain tissue?
A. 1 B. 2
C. 1 while the current increases in the stimulating coil and 2 while the current decreases
D. 2 while the current increases in the stimulating coil, 1 while the current decreases

Averell Hause

Carnegie Mellon University

Problem 65

The brain tissue at the level of the dotted line may be considered as a series of concentric circles, with each circle behaving independently. Where will the induced EMF be the greatest?
A. At the center of the dotted line
B. At the periphery of the dotted line
C. The EMF will be the same in all concentric circles
D. At the center during the increasing phase of the stimulating current and at the periphery during the decreasing phase

Daniel Matthias

Daniel Matthias

Numerade Educator